I started rendering a new animation after completely rendering one before. How do I retrieve that older render?
If you wrote over other files there isn't much you can do other than doing some kind of forensics and trying to recover your files with Data Recovery software (and that is way beyond the scope of this site).

On the other hand there are a couple of things you should do to prevent this from happening again:

-. Uncheck the Overwrite box on the output tab. 



-. Don't use an animation (AVI, Qucktime, h264, Xvid, MPEG) format as output.

-. Render your animation as a series of images (PNG, Tiff, DPX, OpenEXR) and compile as video once you have rendered all your frames. This way, if the computer crashes you'll only loose the current frame but will keep all of the frames that have been rendered already.

-. Save different image sequence/animation renders to different folders under the Output file-path. This will let you save newly rendered sequences while preserving older ones for later analysis.